On the average number of

LXXXIV.1 (1998)

On the average number of

unitary factors of finite abelian groups

by

J. Wu (Nancy)

1. Introduction. Let X be the semigroup of all finite abelian groups with respect to the direct product ⊗ and let E

be the identity of X. For G ∈ X and H ∈ X, we use (G, H) to denote the group of maximal order in X which is simultaneously a direct factor of G and H. We say that G and H are relatively prime if (G, H) = E

. A direct factor D of G is called unitary if D ⊗ E = G and (D, E) = E

. The number of unitary factors of G is denoted by t(G). In 1960, Cohen [2] proved

(1.1) X

|G|≤x

t(G) = A

x log x + A

x + O( √

x log x),

where the summation is over all G in X of order |G| ≤ x and the A

are some effective constants. After a study on Dirichlet’s series associated with t(G), Kr¨atzel [7] found a connection between (1.1) and the following three- dimensional divisor problem:

(1.2) X

n₁n₂n²₃≤x

1 = B

x log x + B

x + B

√

x + ∆(1, 1, 2; x),

where the B

are some effective constants and ∆(1, 1, 2; x) is an error term. Using exponential sum techniques, he showed that ∆(1, 1, 2; x)  x

(log x)

, which implies

(1.3) X

|G|≤x

t(G) = A

x log x + A

x + A

√ x + ∆(x)

with ∆(x)  x

(log x)

. This estimate was improved to ∆(x)  x

(log x)

by Schmidt [11], then to ∆(x) 

x

by Liu [9] and to

∆(x) 

x

by Liu [10], where ε denotes an arbitrarily small positive number.

1991 Mathematics Subject Classification: 11L07, 11N45.

[17]

In this paper, we give a better bound.

Theorem 1. For any ε > 0, we have

∆(1, 1, 2; x) 

x

and ∆(x) 

x

.

For comparison, we have 29/80 = 0.3625 and 47/131 ≈ 0.3587. From [10], we know that in the proof of Theorem 1 the most difficult part is to estimate the exponential sums of type

(1.4) X

h∼H

a

X

n₁∼N₁

X

n₂∼N₂

e(xh/(n

n

)),

where |a

| ≤ 1, e(t) := e

and the notation h ∼ H means cH < h ≤ c

H with some positive unspecified constants c, c

. Liu [10] has treated (1.4) by combining Fouvry–Iwaniec’s method [3] and Kolesnik’s method [6].

We notice that via van der Corput’s B-process the sum (1.4) can be transformed into bilinear exponential sums of type I,

T (M, N ) := X

m∼M

X

n∈I(m)

ϕ

e



X m

n

M

N

 ,

where I(m) is a subinterval of [N, 2N ]. Using the classical A, B process and the well known AB theorem of Kolesnik (see Theorem 1 of [6] and Lemma 1.5 of [8]) we shall prove an estimate for T (M, N ) (see Theorem 3 below). In addition we also use an idea of Jia ([5], Lemma 13) and Liu ([8], Lemma 2.4) to investigate bilinear exponential sums of type II,

S(M, N ) := X

m∼M

X

n∼N

ϕ

ψ

e



X m

n

M

N

 .

Baker and Harman have simplified Jia–Liu’s argument to obtain a slightly more general estimate for S(M, N ) (see Theorem 2 of [1]) than those of Jia and Liu. But all such results contain some restrictions on (X, M, N ) and the number of terms is relatively large; this is not convenient in applications.

Our result (see Theorem 2 below) essentially has the same power as their estimates, but it is without restriction, more general and simpler in form.

Finally, it is worth indicating that we also need Theorem 7 of [12] and Lemma 2.3 of [13] for the proof of Theorem 1.

2. Estimates for exponential sums. We first prove two estimates for S = S(M, N ), defined as in Section 1. In the sequel, the letter ε

denotes a suitably small positive number (depending on α, β and α

at most).

Theorem 2. Let α, β ∈ R with αβ(α − 1)(β − 1) 6= 0, X > 0, M ≥ 1,

N ≥ 1, |ϕ

| ≤ 1, |ψ

| ≤ 1 and L := log(2 + XM N ). If (κ, λ) is an exponent

pair , then

S(M, N )  {(X

M

N

)

(2.1)

+ X

M

N

+ (XM

N

)

+ (XM

N

)

+ M

N

+ M N

+ (X

M

N

)

+ X

M N }L

, S(M, N )  {(XM

N

)

+ (X

M

N

)

+ (XM

N

)

(2.2)

+ M N

+ (X

M

N

)

+ X

M N }L

. P r o o f. We begin in the same way as Jia [5], Liu [8], Baker and Harman [1]. Without loss of generality, we suppose that β > 0 and L is sufficiently large. Let Q ∈ [L, N/L] be a parameter to be chosen later. By Cauchy’s inequality and a “Weyl shift” ([4], Lemma 2.5), we have

|S|

 (M N )

Q + M

N

Q

X

1≤|q1|<Q



1 − |q

| Q

 X

n+q1,n∼N

ψ

ψ

X

m∼M

m

e(Am

t),

where t = t(n, q

) := (n+q

)

−n

and A := X/(M

N

). Splitting the range of q

into dyadic intervals and removing 1 − q

/Q by partial summation, we get

(2.3) |S|

 (M N )

Q

+ LM

N Q

max

1≤Q1≤Q

|S(Q

)|, where

S(Q

) := X

q₁∼Q₁

X

n+q₁,n∼N

ψ

ψ

X

m∼M

m

e(Am

t).

If X(M N )

Q

≥ ε

, by Lemma 2.2 of [12] we can transform the inner- most sum to a sum over l and then using Lemma 2.3 of [12] with n = m we can estimate the corresponding error term. As a result, we obtain

S(Q

)  X

q1∼Q1

X

n+q1,n∼N

ψ

ψ

X

l∈I

l

e(e α(At)

l

) + {(XM

N

Q

)

+ M

N Q

+ (X

M N Q

)

+ (X

M N

)

}L,

where γ := 1/(1−α), e α = |1−α|·|α|

, I := [c

AM

|t|, c

AM

|t|]

and c

= c

(α) are some constants. Exchanging the order of summation and

estimating the sum over l trivially, we find, for some l  X(M N )

Q

, the

inequality

S(Q

)  (XM

N

Q

)

 X X

(n,q₁)∈D₁(l)

ψ

ψ

e(e α(At)

l

)    (2.4)

+ {(XM

N

Q

)

+ M

N Q

+ (X

M N Q

)

+ (X

M N

)

}L,

where D

(l) is a suitable subregion of {(n, q

) : n ∼ N, q

∼ Q

}. Let S

(Q

) be the double sums on the right-hand side of (2.4). Using Lem- ma 2.6 of [12] to relax the range of q

, we see that there exists a real number θ independent of (n, q

) such that

S

(Q

)  L X

n∼N

 X

q₁∼Q₁

ψ

e(θq

)e(e α(At)

l

)   .

If L ≤ Q

≤ Q, using again Cauchy’s inequality and a “Weyl shift” with Q

≤ ε

√

Q

yields

|S

(Q

)/L|

 (N Q

)

Q

+ N Q

Q

X

1≤q₂≤Q₂

|S

(q

, q

)|, where

S

(q

, q

) := X

n∼N

X

q1+q2,q1∼Q1

ψ

ψ

e(t

(n, q

, q

))

and t

(n, q

, q

) := e αA

l

{t(n, q

+ q

)

− t(n, q

)

}. Writing n

:= n + q

, exchanging the order of summation and using Lemma 2.6 of [12], we can deduce

S

(q

, q

) = X X

(n⁰,q1)∈D2

ψ

ψ

e(t

(n

− q

, q

, q

))

 L X

n⁰∼N

 X

q₁∼Q₁

e(θ

q

)e(T (n

, q

, q

))   ,

where T (n

, q

, q

) := t

(n

−q

, q

, q

), D

is a suitable subregion of {(n

, q

) : n

∼ N, q

∼ Q

} and θ

is a real number independent of (n

, q

). A final application of Cauchy’s inequality and a “Weyl shift” with Q

= Q

gives

|S

(q

, q

)/L|

 (N Q

)

Q

+ N Q

Q

X

1≤q₃≤Q₃

X

q₁∼Q₁

|S

(q

, q

, q

)|, where S

(q

, q

, q

) := P

n⁰∼N

e(f (n

)) and f (n

) := T (n

, q

, q

) − T (n

, q

+ q

, q

). It is easy to show that f (n

) satisfies the conditions of exponent pair and f

(n

)  XN

Q

q

q

(n

∼ N ). Hence we have

S

(q

, q

, q

)  (XN

Q

q

q

)

N

+ (XN

Q

q

q

)

,

which implies

S

(Q

)  {(X

N

Q

Q

)

+ N Q

Q

+ (X

N

Q

Q

)

}L

provided Q

≥ L, Q

≤ ε

√

Q

. By Lemma 2.4(ii) of [12] optimizing Q

over (0, ε

√ Q

] yields

S

(Q

)  {(X

N

Q

)

+ N

Q

+ N Q

+ (X

N

Q

)

}L

provided Q

≥ L. In view of the term N Q

L

, this inequality holds trivially when Q

≤ L. Inserting the preceding estimate in (2.4) yields, for any Q

∈ [1, Q],

S(Q

)  {(X

M

N

Q

)

+ X

M

N

Q

+ (X

M

N

Q

)

+ (X

M

N

Q

)

+ (XM

N

Q

)

+ M

N Q

+ (X

M N Q

)

+ (X

M N

)

}L

=: (E

+ E

+ . . . + E

)L

.

Since E

≤ E

and E

= (E

E

)

(M

Q

)

, both E

and E

are superfluous. Replacing Q

by Q and inserting the bound obtained in (2.3), we find, for any Q ∈ [L, N/L],

S  {(X

M

N

Q

)

(2.5)

+ X

M

N

Q

+ (X

M

N

Q)

+ (X

M

N

Q

)

+ M N Q

+ (X

M

N

Q

)

}L

,

where we have used the fact that (X

M

N

Q

)

can be absorbed by M N Q

. In view of M N Q

, the preceding estimate holds trivially when Q ∈ (0, L].

If X(M N )

Q

≤ ε

, we can remove m

by partial summation and then estimate the sum over m by Kuz’min–Landau’s inequality ([4], The- orem 2.1). Hence we see that (2.5) always holds for 0 < Q ≤ N/L. Using Lemma 2.4(ii) of [12] to optimize Q over (0, N/L] yields

S  {(X

M

N

)

+ (X

M

N

)

+ X

M

N

+ M

N

+ (XM

N

)

+ (XM

N

)

+ (XM

N

)

+ (X

M

N

)

+ M N

+ X

M N }L

=: (F

+ F

+ . . . + F

)L

. Since

F

= (F

F

)

N

and F

= (F

F

)

M

, they are both superfluous. This proves (2.1).

To prove (2.2), we take Q

= ε

min{ √

Q

, (X

N

Q

)

} such that

|f

(n

)| ≤ 1/2 for n

∼ N . Thus Kuz’min–Landau’s inequality gives S

(q

, q

, q

)  (XN

Q

q

q

)

, from which we can deduce, as before, the following inequality:

S  {(XM

N

)

+ (XM

N

)

+ (X

M

N

)

+ (X

M

N

)

+ (XM

N

)

+ (X

M

N

)

+ M N

+ X

M N + M

N }L

=: (G

+ G

+ . . . + G

)L

.

It is not difficult to verify that G

= (G

G

)

M

, G

= (G

G

)

(M

N

)

, G

= (G

G

)

M

. Thus G

, G

, G

are superfluous. This completes the proof.

For T = T (M, N ) defined as in Section 1, we have the following result.

Theorem 3. Let α, β ∈ R with αβ(α−1)(β−1)(α+β−1)(2α+β−2) 6= 0, X > 0, M ≥ 1, N ≥ 1, L := log(2 + XM N ), |ϕ

| ≤ 1 and I(m) be a subinterval of [N, 2N ]. Then

T (M, N )  {(X

M

N

)

+ (X

M

N

)

+ (XM

N

)

+ (X

M

N

)

+ (XM

N

)

+ (X

M

N

)

+ (XM

N

)

+ M N

+ X

M N }L

.

P r o o f. If X ≤ ε

N , then T  X

M N by Kuz’min–Landau’s inequal- ity. When X ≥ ε

N , using (2.3) with ψ

= 1, we have, for any 1 ≤ Q ≤ ε

N , (2.6) |T |

 (M N )

Q

+ LM

N Q

max

1≤Q₁≤Q

|T (Q

)|, where

T (Q

) := X

q∼Q1

X

n∈I₁(q)

X

m∈J(n,q)

m

e



X m

t(n, q) M

N

 ,

t(n, q) := (n + q)

− n

, I

(q) is a subinterval of [N, 2N ] and J(n, q) a

subinterval of [M, 2M ].

If L := X(M N )

Q

≥ ε

, similarly to (2.4), we can prove, for some l  L,

T (Q

)  (XM

N

Q

)

 X X

(n,q)∈D(l)

e(f (n, q))    + {M

N Q

+ (XM

N

Q

)

+ (X

M N

)

+ (X

M N Q

)

}L,

where f (n, q) := e α(XQ

/N )(l/L)

{t(n, q)/(N

Q

)}

and D(l) is a suitable subregion of {(n, q) : n ∼ N, q ∼ Q

}. It is easy to show that f (n, q) satisfies the condition of Lemma 1.5 of [8] (which is a revised form of Theorem 1 of Kolesnik [6]) with A = XN

Q

/(N

Q

)

,

∆ = Q

/N . By this lemma with (F, X, Y ) = (XN

Q

, N, Q

), we obtain the estimate

T (Q

)  {(X

M

N

Q

)

+ (X

M

N

Q

)

(2.7)

+ (XM

N Q

)

+ (X

M

N

Q

)

+ (XM

N

Q

)

+ (X

M

N

Q

)

+ (X

M

Q

)

+ (X

M N

)

+ (X

M N Q

)

}L

, where we have used the fact that M

N Q

+ (XM

N

Q

)

can be absorbed by (XM

N

Q

)

+ (X

M

Q

)

(in view of the hypothesis X ≥ ε

N ).

If L ≤ ε

, the Kuz’min–Landau inequality implies that (2.7) also holds.

Replacing Q

by Q and inserting into (2.6) yield

|T |

 {(X

M

N

Q

)

+ (X

M

N

Q

)

+ (XM

N

)

+ (X

M

N

Q

)

+ (XM

N

Q)

+ (X

M

N

Q

)

+ (X

M

N

Q

)

+ (M N )

Q

}L

,

where we have eliminated two superfluous terms X

M

N

Q

and (X

M

N

Q

)

(which can be absorbed by (M N )

Q

). Using Lem- ma 2.4(ii) of [12] to optimize Q over (0, ε

N ] gives the required result. This concludes the proof.

Next we shall apply Theorems 2 and 3 to treat S

:= X

m₁∼M₁

X

m₂∼M₂

X

m₃∼M₃

ψ

e



X m

m

m

M

M

M

 ,

S

:= X

m₁∼M₁

X

m₂∼M₂

X

m₃∼M₃

ϕ

ψ

e



X m

m

m

M

M

M

 ,

which are general forms of (1.4). The following results will be used in the

proof of Theorem 1.

Corollary 1. Let α

∈ R with α

α

(α

+1)(α

−jα

−j) 6= 0 (j = 1, 2), X > 0, M

≥ 1, |ϕ

| ≤ 1, |ψ

| ≤ 1 and let Y := 2 + XM

M

M

. If (κ, λ) is an exponent pair , then for any ε > 0,

S

 {(X

M

M

M

)

+ X

M

1

M

M

+ (X

M

M

M

)

+ (X

M

M

M

)

+ X

M

1

M

M

+ (XM

M

)

+ (X

M

M

M

)

+ M

M

+ X

M

M

+ X

M

M

M

}Y

. In particular , if X ≥ M

≥ M

, then

S

 {(X

M

M

M

)

(2.8)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (XM

M

)

}Y

,

S

 {(X

M

M

M

)

+ (X

M

M

M

)

(2.9)

+ (X

M

M

)

+ (XM

M

M

)

}Y

.

P r o o f. If M

:= X/M

≤ ε

, the Kuz’min–Landau inequality implies S

 X

M

M

M

.

Next we suppose M

≥ ε

. As before, using Lemma 2.2 of [12] to the sum over m

and estimating the corresponding error term by Lemma 2.3 there with n = m

, we obtain

S

 X

M

S+(X

M

+M

M

+X

M

M

+X

M

M

M

) log Y, where

S := X

m₁∼M₁

X

m₂∼M₂

X

m⁰₃∼M₃⁰

e

ϕ

ψ e

ξ

3

e

 e

α

X m

m

m

M

M

M



= X

m₁∼M₁

X

m⁰₂∼M₂⁰

e

ϕ

ξ e

e

 e

α

X m

m

M

M

 ,

and β

:= α

/(1+α

) (j = 1, 2), e α

:= |1+α

|·|α

|

, | e ϕ

| ≤ 1, | e ψ

| ≤ 1,

|ξ

3

| ≤ 1, M

:= M

M

, e ξ

2

:= P P

m2m⁰₃=m⁰₂

ψ e

ξ

3

. By Theorem 2 with (M, N ) = (M

, M

) we estimate S to get the first assertion.

In particular taking (κ, λ) = BA

,

=

,

yields

S

 {(X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (XM

M

)

+ (X

M

M

M

)

+ M

M

+ X

M

M

+ X

M

M

M

}Y

=: (H

+ H

+ . . . + H

)Y

.

Since X ≥ M

≥ M

, we have H

≤ H

, H

≤ H

, H

≤ H

(8 ≤ j ≤ 10) and thus H

, H

(7 ≤ j ≤ 10) are superfluous. This proves (2.8).

The last inequality can be proved similarly by using Theorem 7 of [12]

with (M

, M

, M

) = (M

, 1, M

). This completes the proof.

Corollary 2. Let α

∈ R with α

α

(α

−1)(α

−2)(α

+1)(α

−α

−1) 6= 0, X > 0, M

≥ 1, |ψ

| ≤ 1 and let Y := 2 + XM

M

M

. If M

≥ M

, then for any ε > 0 we have

S

 {(X

M

M

M

)

+ (X

M

M

M

)

(2.10)

+ (XM

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (XM

M

)

+ X

M

M

M

}Y

, S

 {(X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

)

(2.11)

+ (X

M

M

)

+ M

M

+ X

M

M

M

}(log 2Y )

. P r o o f. As before we may suppose M

:= X/M

≥ ε

and prove

S

 X

M

T (2.12)

+ (X

M

+ M

M

+ X

M

M

+ X

M

M

M

) log Y, where

T := X

m1∼M1

X

m2∼M2

X

m⁰₃∈I₃

g(m

) e ψ

ξ

3

e

 e

α

X m

m

m

M

M

M

 ,

I

:= [c

(m

/M

)

(m

/M

)

M

, c

(m

/M

)

(m

/M

)

M

], and β

, e α

, e ψ

, ξ

3

are defined as before, c

= c

(α

) are constants, g(m

) is a monomial with |g(m

)| ≤ 1. We define e ξ

2

and M

in the same way as in the proof of Corollary 1. Exchanging the order of summation, we have

T = X

m⁰₂∼M₂⁰

ξ e

2

X

m1∈I1(m⁰₂)

g(m

)e

 e

α

X m

m

M

M

 ,

where I

(m

) is a subinterval of [M

, 2M

]. Removing g(m

) by partial

summation and estimating the double sum obtained by Theorem 3 with

(M, N ) = (M

, M

), we find

T  {(X

M

M

)

+ (X

M

M

)

+ (XM

M

)

+ (X

M

M

)

+ (XM

M

)

+ (X

M

M

)

+ (XM

M

)

+ M

M

}Y

.

Inserting into (2.12) and noticing that the last four terms on the right- hand side of (2.12) can be absorbed by (XM

M

)

, we obtain (2.10). The inequality (2.11) is (2.7) of [13] with (M

, M

, M

) = (M

, M

, M

) and (α

, α

, α

) = (α

, α

, −α

). This concludes the proof.

3. Proof of Theorem 1. We shall prove only (3.1) ∆(1, 1, 2; x) 

x

,

since this implies ∆(x) 

x

by a simple convolution argument. For this we recall some standard notations. Let u := (u

, u

, u

) be a permuta- tion of (1, 1, 2) and let N := (N

, N

) ∈ N

. We write ψ(t) := {t} − 1/2 ({t}

is the fractional part of t) and define S(u, N; x) := X

1

ψ((x/(n

n

))

), where the summation condition of P

1

is n

n

≤ x, n

(≤)n

, n

∼ N

, n

∼ N

. The notation n

(≤)n

means that n

= n

for u

< u

, and n

< n

otherwise. It is well known that for proving (3.1) it suffices to verify

S(u, N; x)  x

for u = (1, 1, 2), (2, 1, 1), (1, 2, 1).

Since S(1, 1, 2, N; x)  x

(see [10], p. 263), it remains to consider u = (2, 1, 1), (1, 2, 1). We shall prove the desired estimate for u = (2, 1, 1) in two cases according to the size of N

, which we shall formulate as two lemmas. The case of u = (1, 2, 1) can be treated similarly (more easily).

We recall that we have N

≤ N

≤ G := x/(N

N

), N

N

≤ x

when u = (2, 1, 1). This fact will be used (implicitly) many times in the proofs of Lemmas 3.1 and 3.2.

Lemma 3.1. For u = (2, 1, 1), we have

S(u, N; x) 

{(x

N

)

+ (xN

)

+ (x

N

)

+ x

}x

. In particular , if N

≤ x

, then S(u, N; x) 

x

.

P r o o f. By Lemma 2.5 of [12], we have, for any H ≥ 1, (3.2) S(u, N; x)  H

N

N

+ (log x) max

1≤H₀≤H

H

|S(H

, N)|, where

S(H

, N) := X

h∼H0

a

X

n1∼N1

X

n2∼N2

e(hx/(n

n

)), |a

| ≤ 1.

The inequalities (2.8) and (2.9) with (X, M

, M

, M

) = (GH

, N

, H

, N

) imply

S(H

, N)  {(G

N

N

)

+ (GN

H

)

+ χ

+ χ

}H

x

, S(H

, N)  {(G

N

N

H

)

+ (G

N

N

H

)

+ (G

N

H

)

+ (xN

)

}H

x

=: {D

+ D

+ D

+ (xN

)

}H

x

,

with χ

:= (G

N

N

H

)

and χ

:= (G

N

N

H

)

, where we have used the fact that (G

N

N

)

≤ (G

N

N

)

(in view of N

≤ x

). From these, we deduce that for any H

≥ 1,

S(H

, N)  n

(G

N

N

)

+ (GN

H

)

+ (xN

)

+ X

1≤j≤2

X

1≤k≤3

R

o

H

x

,

where R

:= min{χ

, D

}. Since

 



 

R

≤ (χ

D

)

= (x

N

)

, R

≤ (χ

D

)

= x

,

R

≤ (χ

D

)

= (x

N

)

,

 



 

R

≤ (χ

D

)

= (x

N

)

< x

, R

≤ (χ

D

)

= (x

N

)

< x

, R

≤ (χ

D

)

= (x

N

)

< x

, and (x

N

)

≤ (x

N

)

, we have

S(H

, N)  {(G

N

N

)

+ (GN

H

)

+ (xN

)

+ (x

N

)

+ x

}H

x

.

Inserting into (3.2) and optimizing H by Lemma 2.4(iii) of [12] yield the desired estimate.

Lemma 3.2. For u = (2, 1, 1), we have

S(u, N; x)  {(x

/N

)

+ (x

/N

)

+ (x

/N

)

+ (x

/N

)

+ (x

/N

)

+ x

}x

. In particular , if N

≥ x

, then S(u, N; x) 

x

.

P r o o f. Corollary 2 with (X, M

, M

, M

) = (GH

, N

, H

, N

) gives S(H

, N)  {L(H

) + (G

N

N

H

)

+ (GN

N

H

)

(3.3)

+ (G

N

N

H

)

+ (G

N

N

H

)

}H

x

=: {L(H

) + σ

+ σ

+ σ

+ σ

}H

x

,